Equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces

TL;DR

This paper determines the minimal area of certain hyperbolic surfaces tiled by regular triangles and characterizes all cases where equality holds in Fejes Toth's triangle bound for these surfaces.

Contribution

It provides the first complete characterization of equality cases in Fejes Toth's triangle bound for hyperbolic surfaces, including minimal area surfaces for k>6.

Findings

Minimal area hyperbolic surfaces tiled by regular triangles for k>6

Complete description of equality cases in Fejes Toth's triangle bound

Identification of surfaces achieving the bound

Abstract

For k>6, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle 2\pi/k. Based on this, all the cases of equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces are described.